Homework 1 Solution

Description

1. Plot the following sinusoids accurately. Label both axes and the critical values of the signal such as the minimum and maximum amplitude, period, zero crossings, etc. Time units are in seconds and the arguments are in radians. Include about one period to the left and two periods to the right of the origin.

1. 1. x(t) = 5 sin(6πt − ^π₃ )

1. 2. x(t) = −6 cos(5t + 2)

2. x(t) and y(t) are two sinusoidal signals defined as follows:

√		π				π
x(t) = 3	2 cos(2t −			)	y(t) = 3 sin(2t +		)
			3			4

1. 1. Find the phasors X and Y representing the sinusoids x(t) and y(t), respectively.

1. 2. Find x(t) + y(t) using the phasor addition technique and verify your result by providing a phasor addition diagram, drawn to scale.

3. Determine whether the following signals are periodic or not. If periodic, iden-tify the fundamental frequency and the period and indicate which harmonics exist in the frequency spectrum of the signal.

3. 1. x(t) = 3 sin(3t) cos( ^π₅ t + ^π₃ ) − 1

b) x(t) = 7 sin(√
	3t + 5) + 2 cos(πt)
c) x(t) = cos(12t) − 3 cos(16t)
d) x(t) = sin2(60√					√		t +	π	)
			2	t) + cos2(24		2
						4

4. A continuous-time signal is real valued and has a fundamental period of T_o = 5 sec. The non-zero Fourier series coefficients of x(t) in the interval 0 < k < ∞ of the synthesis equation are specified as follows:

a₀ = ₋4, a₁ = ₋j a₅ = ₋4e^{−j π}3 a₈ = 2e^{j π}3

Express x(t) in the form:

∞

x(t) = A₀ + A_k cos(ω_kt + φ_k)

k=1

5. Find all the Fourier series coefficient of the following signal without having to evaluate the analysis integral:

x(t) = 5 − 2e−^jt + 3 sin(−t + π/4) − 4 cos(3t + 5) + 3 cos(4t) cos(5t + π/2)

1. What is the average value of x(t)?

2. What is the fundamental period of x(t)?

3. Is x(t) a real signal?

4. Plot the frequency spectrum of x(t). Label all frequencies and complex amplitudes. Which harmonics exist in the spectrum?

6) If a continuous-time periodic signal has the Fourier series coefficients a_k, k = 0, ±1, ±2, ±3, . . ., derive the Fourier series coefficients of the following sig-nals in terms of a_k:

a) y(t) = cx(t − t_o) where c and t_o are constants b) y(t) = ^dx(t)

dt

7. Find all of the Fourier series coefficients of a full-wave rectified sine waveform with amplitude 5 units and radian frequency ω_o = 10π rad/sec. The phase shift of the waveform is zero.

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